Dirac Delta Function

Date: 05/15/2000 at 19:54:08
From: Betty
Subject: Dirac's delta function
How is one to interpret Dirac's delta function? What is it, and what
does it mean? It is used in proofs in a text on stochastic
differential equations I'm reading, and I can't find a description of
what it represents in any of the texts available to me. (I live in a
very rural area, and have to rely on my own library.)

Date: 05/15/2000 at 23:05:42
From: Doctor Douglas
Subject: Re: Dirac's delta function
Hi Betty,
Thanks for writing to Ask Dr. Math.
The Dirac delta function is a special kind of object - it is not a
function in the usual sense, although we sometimes think of it that
way. Basically, it can be thought of as the "function" d(x) such that:
d(x) = 0 for x <> 0
= Infinity for x = 0
and the "size" of the infinity is such that the integral of d(x) with
respect to x from a to b is zero if a and b are of the same sign, and
the integral is exactly unity if a < 0 < b. More precisely, it can be
thought of as the limit of a series of functions that get more and
more peaked near zero, and whose area is always unity.
It is always to be implicitly used inside an integral, even though we
carry out operations such as:
f(x) = d(x) + d(x-2) (when integrated, this has area 2)
You can read more about the Dirac delta function from Eric Weisstein's
MathWorld site here:
Delta Function
http://mathworld.wolfram.com/DeltaFunction.html
If you have more questions, please feel free to write back!
- Doctor Douglas, The Math Forum
http://mathforum.org/dr.math/